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Some asymptotic laws for crossings and excursions. (English) Zbl 0666.60070

Les processus stochastiques, Coll. Paul Lévy, Palaiseau/Fr. 1987, Astérisque 157-158, 59-74 (1988).
[For the entire collection see Zbl 0649.00017.]
Working with a recurrent Hunt process having an invariant reference measure, the authors consider various processes associated with a pair \(A, C\) of closed subsets of the state space such that \(A\cap C\) is polar, or with \(M\), a closed optimal subset of \((0,\infty)\). For the counting processes for the number of “crossings” from \(A\) to \(C\) in \((0,t]\), or for the number of excursions outside of \(M\) of some fixed type in \((0,t]\), general asymptotic results are deduced from an ergodic theorem for additive processes.
The Markov chains of starting points of crossings or of excursions of a certain type are described. Asymptotics for crossings of planar Brownian motion, both in case the logarithmic capacity of \(A\) relative to \(C\) is finite (e.g., \(A\) and \(C\) are non-intersecting circles) and in case it is infinite (as when \(A\cap C\) consists of a finite number of points) are derived. The analysis for excursions is related to the notion of Palm measure associated with a stationary process of excursions.

MSC:

60J25 Continuous-time Markov processes on general state spaces
60J55 Local time and additive functionals
60J65 Brownian motion

Citations:

Zbl 0649.00017